Spacecraft trajectory design is a science that requires high precision with little error. One of the most classic trajectory design problems is the restricted three-body problem. Two methods to develop the trajectory of a spacecraft under the influence of two celestial bodies are through the use of the equations of motion, and the patched-conic approximation. Popular tools such as MATLAB can be used to solve the equations of motion if great care is taken when selecting an ODE solver since the results are dramatically different between different solvers. As a result, these tools aren’t very robust and can create significant errors, so a different approach must be used for generalized scenarios when an exact solution for comparison is unavailable. The patched-conic approximation can be easily used in a program such as MATLAB, but its exclusion of one of the two celestial bodies at every point in the trajectory creates drawbacks and significant errors.
To avoid the errors that exist when using the patched-conic approach, research was put into the development of a simple model that could propagate a spacecraft’s trajectory under the effect of two celestial bodies while being robust enough to code and solve in a widely available program such as MATLAB. This model acts as a modification to the patched-conic approach. Throughout the trajectory the effect of the primary celestial body of the system on the spacecraft was calculated, as in the patched-conic approach, however unlike the patched-conic approach this effect is not ignored when the spacecraft reaches the secondary body’s sphere of influence. Furthermore, the effect of the secondary body was also considered even when the spacecraft is outside the secondary body’s sphere of influence. Then, by applying a weighted average to the spacecraft’s radius and velocity components respective to each celestial body, an updated state would be created that would allow the model to accurately propagate the trajectory. This would be compared to a numerically generated ‘exact’ solution to determine the errors.
Algorithms that propagate the spacecraft’s trajectory out with respect to both celestial bodies were created and tested, including the propagation of the secondary celestial body’s orbit itself. A scheme based on the geometry was used in an attempt to combine the spacecraft’s states with respect to both celestial bodies using a weighted average. This scheme was tested at multiple points throughout the trajectory using a variety of weights, but no attempts were met with any success. However, the routines propagating the trajectories of the celestial bodies and spacecraft were proven to work correctly, and an initial foundation in creating a scheme to combine the spacecraft’s state has been laid out.